Source parameters for historical earthquakes worldwide are compiled to develop a series of empirical relationships among moment magnitude ( M ), surface rupture length, subsurface rupture length, downdip rupture width, rupture area, and maximum and average displacement per event. The resulting data base is a significant update of previous compilations and includes the additional source parameters of seismic moment, moment magnitude, subsurface rupture length, downdip rupture width, and average surface displacement. Each source parameter is classified as reliable or unreliable, based on our evaluation of the accuracy of individual values. Only the reliable source parameters are used in the final analyses. In comparing source parameters, we note the following trends: (1) Generally, the length of rupture at the surface is equal to 75% of the subsurface rupture length; however, the ratio of surface rupture length to subsurface rupture length increases with magnitude; (2) the average surface displacement per event is about one-half the maximum surface displacement per event; and (3) the average subsurface displacement on the fault plane is less than the maximum surface displacement but more than the average surface displacement. Thus, for most earthquakes in this data base, slip on the fault plane at seismogenic depths is manifested by similar displacements at the surface. Log-linear regressions between earthquake magnitude and surface rupture length, subsurface rupture length, and rupture area are especially well correlated, showing standard deviations of 0.25 to 0.35 magnitude units. Most relationships are not statistically different (at a 95% significance level) as a function of the style of faulting: thus, we consider the regressions for all slip types to be appropriate for most applications. Regressions between magnitude and displacement, magnitude and rupture width, and between displacement and rupture length are less well correlated and have larger standard deviation than regressions between magnitude and length or area. The large number of data points in most of these regressions and their statistical stability suggest that they are unlikely to change significantly in response to additional data. Separating the data according to extensional and compressional tectonic environments neither provides statistically different results nor improves the statistical significance of the regressions. Regressions for cases in which earthquake magnitude is either the independent or the dependent parameter can be used to estimate maximum earthquake magnitudes both for surface faults and for subsurface seismic sources such as blind faults, and to estimate the expected surface displacement along a fault for a given size earthquake.

New Empirical Relationships among Magnitude, Rupture Length, Rupture Width, Rupture Area, and Surface Displacement

An earthquake model is derived by considering the effective stress available to accelerate the sides of the fault. The model describes near- and far-field displacement-time functions and spectra and includes the effect of fractional stress drop. It successfully explains the near- and far-field spectra observed for earthquakes and indicates that effective stresses are of the order of 100 bars. For this stress, the estimated upper limit of near-fault particle velocity is 100 cm/sec, and the estimated upper limit for accelerations is approximately 2g at 10 Hz and proportionally lower for lower frequencies. The near field displacement u is approximately given by u(t) = (σ/μ) βr(1 - e−t/r) where. σ is the effective stress, μ is the rigidity, β is the shear wave velocity, and τ is of the order of the dimension of the fault divided by the shear-wave velocity. The corresponding spectrum is

Ω(ω)=σβμ1ω(ω2+τ−2)1/2(1)

The rms average far-field spectrum is given by

〈 Ω(ω) 〉=〈 Rθϕ 〉σβμrRF(e)1ω2+α2(2)

where 〈Rθϕ〉 is the rms average of the radiation pattern; r is the radius of an equivalent circular dislocation surface; R is the distance; F(e) = {[2 – 2e][1 – cos (1.21 eω/α)] +e2}1/2; e is the fraction of stress drop; and α = 2.21 β/r. The rms spectrum falls off as (ω/α)−2 at very high frequencies. For values of ω/α between 1 and 10 the rms spectrum falls off as (ω/α)−1 for e < ∼0.1. At low frequencies the spectrum reduces to the spectrum for a double-couple point source of appropriate moment. Effective stress, stress drop and source dimensions may be estimated by comparing observed seismic spectra with the theoretical spectra.

http://www.openseismo.org/contributors/Lee/MoWorking_Backups/Mo2012_0424backup/MoWorking/Papers_NotUsed/0_Theory/928_Bru70_Brune_JGR1970_p4997.pdf

Tectonic stress and the spectra of seismic shear waves from earthquakes

This essential reference for graduate students and researchers provides a unified treatment of earthquakes and faulting as two aspects of brittle tectonics at different timescales. The intimate connection between the two is manifested in their scaling laws and populations, which evolve from fracture growth and interactions between fractures. The connection between faults and the seismicity generated is governed by the rate and state dependent friction laws - producing distinctive seismic styles of faulting and a gamut of earthquake phenomena including aftershocks, afterslip, earthquake triggering, and slow slip events. The third edition of this classic treatise presents a wealth of new topics and new observations. These include slow earthquake phenomena; friction of phyllosilicates, and at high sliding velocities; fault structures; relative roles of strong and seismogenic versus weak and creeping faults; dynamic triggering of earthquakes; oceanic earthquakes; megathrust earthquakes in subduction zones; deep earthquakes; and new observations of earthquake precursory phenomena.

The Mechanics of Earthquakes and Faulting

The nonlinear inverse problem for seismic reflection data is solved in the acoustic approximation. The method is based on the generalized least‐squares criterion, and it can handle errors in the data set and a priori information on the model. Multiply reflected energy is naturally taken into account, as well as refracted energy or surface waves. The inverse problem can be solved using an iterative algorithm which gives, at each iteration, updated values of bulk modulus, density, and time source function. Each step of the iterative algorithm essentially consists of a forward propagation of the actual sources in the current model and a forward propagation (backward in time) of the data residuals. The correlation at each point of the space of the two fields thus obtained yields the corrections of the bulk modulus and density models. This shows, in particular, that the general solution of the inverse problem can be attained by methods strongly related to the methods of migration of unstacked data, and commerc...

/pdf/inversion-of-seismic-reflection-data-in-the-acoustic-37keqm0xle.pdf

Inversion of seismic reflection data in the acoustic approximation

This paper introduces a method for the evaluation of the seismic risk at the site of an engineering project. The results are in terms of a ground motion parameter (such as peak acceleration) versus average return period. The method incorporates the influence of all potential sources of earthquakes and the average activity rates assigned to them. Arbitrary geographical relationships between the site and potential point, line, or areal sources can be modeled with computational ease. In the range of interest, the derived distributions of maximum annual ground motions are in the form of Type I or Type II extreme value distributions, if the more commonly assumed magnitude distribution and attenuation laws are used.

https://courses.washington.edu/ess502/Cornell_1968.pdf

Engineering seismic risk analysis

In the past decade, seismology has matured as a quantitative science through an extensive interplay between theoretical and experimental workers. Several specialized journals have recorded this progress in thousands of pages of research papers, yet such a forum does not bring out key concepts systematically. Because many graduate students have expressed their need for a textbook on this subject and because many methods of seismogram analysis now used almost routinely by small groups of seismologists have never been adequately explained to the wider audience of scientists and engineers who work in the peripheral areas of seismology, we have here attempted to give a unified treatment of those methods of seismology th at are currently used in interpreting actual data. We develop the theory of seismic-wave propagation in realistic Earth models. We study specialized theories of fracture and rupture propagation as models of an earthquake, and we supplement these theoretical subjects with practical descriptions of how seismographs work and how data are analyzed and inverted. Our text is arranged in two volumes. Volume I gives a systematic development of the theory of seismic-wave propagation in classical Earth models, in which material properties vary only with depth. It concludes with a chapter on seismometry. This volume is intended to be used as a textbook in basic courses for advanced students of seismology. Volume II summarizes progress made in the major frontiers of seismology during the past decade. It covers a range of special subjects, including chapters on data analysis and inversion, on successful methods for quantifying wave propagation in media varying laterally (as well as with depth), and on the kinematic and dynamic aspects of motions near a fault plane undergoing rupture. The second volume may be used as a texbook in graduate courses on tectonophysics, earthquake mechanics, inverse problems in geophysics, and geophysical data processing.n

Quantitative seismology : theory and methods

Coda waves from small local earthquakes are interpreted as backscattering waves from numerous heterogeneities distributed uniformly in the earth's crust. Two extreme models of the wave medium that account for the observations on the coda are proposed. In the single backscattering model the scattering is considered to be a weak process, and the loss of seismic energy by scattering is neglected. In the diffusion model the seismic energy transfer is considered as a diffusion process. Both models lead to similar formulas that allow an accurate separation of the effect of earthquake source from the effects of scattering and attenuation on the coda spectra. A unique difference was found in the scaling law of earthquake source spectra between central California and western Japan, which may be attributed to the difference in inhomogeneity length of the earth's crust. The Q of coda waves in the two regions is strongly frequency dependent with values increasing from 50–200 at 1 Hz to about 1000–2000 at 20 Hz. This observation is interpreted as a combined effect of variation of Q with depth and frequency-dependent composition of coda waves described below. The turbidity coefficient of the lithosphere required at 1 Hz to explain the observed coda as body wave scattering is orders of magnitude greater than previously known values such as those obtained by Aki (1973) and Capon (1974) under the Montana Lasa from the amplitude and phase fluctuations of teleseismic P waves. From the high attenuation and turbidity obtained at this frequency we conclude that at around 1 Hz the coda is made of backscattering surface waves from heterogeneities in the shallow, low-Q lithosphere. The high Q observed for the coda at frequencies higher than 10 Hz, on the other hand, eliminates the possibility that these waves are backscattering surface waves. We conclude that at these high frequencies the coda must be made of backscattering body waves from heterogeneities in the deep lithosphere. The low turbidities found for deep earthquake sources under western Japan are consistent with this model of coda wave generation.

Origin of coda waves: Source, attenuation, and scattering effects

The dependence of the amplitude spectrum of seismic waves on source size is investigated on the basis of two dislocation models of an earthquake source. One of the models (by N. Haskell) is called the ω³ model, and the other, called the ω² model, is constructed by fitting an exponentially decaying function to the autocorrelation function of the dislocation velocity. The number of source parameters is reduced to one by the assumption of similarity. We found that the most convenient parameter for our purpose is the magnitude Ms, defined for surface waves with period of 20 sec. Spectral density curves are determined for given Ms. Comparison of the theoretical curves with observations is made in two different ways. The observed ratios of the spectra of seismic waves with the same propagation path but from earthquakes of different sizes are compared with the corresponding theoretical ratios, thereby eliminating the effect of propagation on the spectrum. The other method is to check the theory with the empirical relation between different magnitude scales defined for different waves at different periods. The ω² model gives a satisfactory agreement with such observations on the assumption of similarity, but the ω³ model does not. We find, however, some indications of departure from similarity. The efficiency of seismic radiation seems to increase with decreasing magnitude if the Gutenberg-Richter magnitude-energy relation is valid. The assumption of similarity implies a constant stress drop independent of source size. A preliminary study of Love waves from the Parkfield earthquake of June 28, 1966, shows that the stress drop at the source of this earthquake is lower than the normal value (around 100 bars) by about 2 orders of magnitude.

Scaling law of seismic spectrum

Quantitative Seismology, 2nd Ed.

/pdf/space-and-time-spectra-of-stationary-stochastic-waves-with-2a99hs28a9.pdf

Keiiti Aki

Papers

Quantitative seismology : theory and methods

Origin of coda waves: Source, attenuation, and scattering effects

Scaling law of seismic spectrum

Quantitative Seismology, 2nd Ed.

Space and Time Spectra of Stationary Stochastic Waves, with Special Reference to Microtremors